You are getting your Consulates/High Commisions and Embassies mixed up. Staff at the prior don't generally get diplomatic privilege/immunity, whilst the latter do.
The rank of Consul is not a diplomatic rank, presumably as a Consulate is not a diplomatic mission. The diplomatic ranks are: Attaché, Third Secretary, Second Secretary, First Secretary, Counsellor, Minister, Ambassador.
British Consulates can be considered a satellite location of the British Embassy of that country and typically do consular work only (visa, passport, British education (British council) etc). They have no relationship to the government of the host country. For example, Hong Kong and Shanghai have a consulate, but they report to the British Embassy in Beijing. The staff in Hong Kong and Shanghai are not diplomats per se, but the staff in Beijing are.
Embassy staff only hold diplomatic rank/status, when overseas in their place of posting ONLY (not in any other country). Every posting is 3-4 years, every third posting is back to London as a standard civil servant.
A word of warning, work in HM Diplomatic Corp is likely not what you think, probably in the same context that MI6 isn't full of James Bond characters. Pay is generally lame, with the best uplifts reserved for the worst countries in the world where you really won't have any lifestyle. The perks aren't what they used to be, you can't send your kids to Hogwarts anymore without coughing up a 33% of the fees yourself!
Rumour has it that climbing the ranks in the FCO is very slow. It is quicker to join the Home Office, climb the civil service ranks, then transfer over to foreign office at a later date, in a much more senior rank than you would have if you joined the FCO from the start.